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I think there is a fallacy here actually. Some of the comments come close to touching it. If there are 3 islanders with blue eyes, each can see 2 blue-eyed peo

I think there is a fallacy here actually. Some of the comments come close to touching it. If there are 3 islanders with blue eyes, each can see 2 blue-eyed people. The information that there is at least 1 blue is therefore not new information, and can't change the outcome. The outsider's comment is superfluous given that N > 3.

alwaysmetara 11y ago
The important point is that the outsider's comment "synchronizes" the islander's information. On Day 2, say you are a blue eyed person. You can see two other blue eyed people. Since these two people don't commit suicide during that day, you know there must be three blue eyed people with one of them being you. So now you know your eye color.
- deleted 11y ago
  [deleted]
- jack9 11y ago
  > you know there must be three blue eyed people Why? The visitor didn't synchronize anything. All of the villagers already knew there was at least 1 blue eyed person. Per the story, 99 or 100 depending on your perspective.
  - alwaysmetara 11y ago
    Everyone commits suicide at noon in view of everyone else. So the lowest possible number of blue eyed people go up one per day. Here's a longer explanation: Since people who know their eye color suicides at noon in plain sight of everyone else, we can set up induction. If there's only one blue eyed person, they would suicide the first day at noon. Since everyone sees > 1 blue eyed people, they know this wouldn't happen. Since this doesn't happen, it's common knowledge that there is more than 1 blue eyed person. So if you only see one other blue eyed person, you must have blue eyes (but you don't know this yet because you see two blue eyed people). On the second day, if there are only two blue eyed people, they would both suicide at noon. You see two blue eyed people. No one commits suicide because they either see two blue eyed people (if they have blue eyes) or three blue eyed people (if they don't). Now, everyone knows that there are more than two blue eyed people. Since no one committed suicide the day before, you reason that there must be three blue eyed people, with the 2 people you see plus yourself. Thus, you know your eye color. This is a bit long but I hope it's clear.
    - chii 11y ago
      the argument i hear about induction is that it's only valid IF n=1 is a true situation, n=2 is a true situation, etc, up to n=k (where k is 100 here). But we are only given n=100, and we cannot say that n=1,2,3... is true. The best counter example is the surprise execution paradox: suppose you are a prisoner, about to be executed next week. However, a strange rule applies - if you are not surprised by the date of your execution, you get to go free. You reason that you cannot be executed on sunday, the last day, since by friday, you'd know, and thus not be surprised. by induction, friday is also not a possible day, since you'd know by thurs, etc. You conclude that you cannot be surprised, and this will not get executed. But come wed, you get executed, and is completely surprised!
      - knughit 11y ago
        This problem is different from the surprise paradox, which relies on probability changing each day, and a possibility that either (a) the executioner will be proven a liar by randomly guessing Sunday, or (b) you confidently guessing Wednesday and being right.
    - jack9 11y ago
      > Now, everyone knows that there are more than two blue eyed people. That's incorrect, in a subtle way. The villagers (every single one) knew that before. It isn't discovered knowledge. Everyone (including the visitor) knows there are 99 or 100 because they can count them already, prior to the visitor. So nobody is suiciding for 99 days. As an individual who sees 99 other blue eyes, why would you think or not think that you have blue eyes on the next day?
      - alwaysmetara 11y ago
        I meant strictly more, as in at least three. If you know that there are more than two blue eyed people and you can only see two, you know that you have blue eyes. If you can see 99 blue eyes, you know that all the other people see either 98 or 99 (if there's a total of 99 blue eyes) or 99 or 100 (if there's 100 blue eyes). However, if some people see 98 and know that there's at least 99 people with blue eyes, then they would all suicide. Since they don't, you know everyone sees either 99 or 100 blue eyes. Since you see 99, you must have blue eyes.
- awesomepantsm 11y ago
  Why can't they synchronize at any other moment? They already knew N is either 99 or 100 even before the visitor came.
  - millstone 11y ago
    Consider the case of 2 blue eyed people, Ben and Betty. Everybody knows there is at least one blue eyed person. But does everybody know that? That is, does everybody know that (everybody knows there is one blue eyed person)? From Ben's perspective, Betty might be the only blue-eyed person, so Ben does not know if Betty knows there is at least one blue eyed person. This is important because Ben needs to correlate Betty's actions with her knowledge; he can't do that if he doesn't know what her knowledge is. After the traveler's announcement, Ben knows that Betty knows there is at least one blue eyed person. This is the new information. And with three people, we add another layer: Ben does not know if Betty knows that Bobby has blue eyes. And so on. The visitor's announcement is special because it is infinitely layered. Everyone knows one person has blue eyes, and everyone knows that everyone knows, and so on, ad infinitum. This is called common knowledge: https://en.wikipedia.org/wiki/Common_knowledge_(logic) https://en.wikipedia.org/wiki/Common_knowledge_(logic)). Though to be fair, this isn't explicitly stated: the traveler "addresses the entire tribe" but it's critical that everyone in the tribe knows this is common knowledge. Merely addressing the entire tribe is not enough: if the traveler's announcement was in the form of a BCC mass mailing, there would be no new information and so no suicides.
    - knughit 11y ago
      It isn't infinitely layered! It is N-layered, which is critical.
  - daxfohl 11y ago
    "synchronize" is a bit vague. It can be made more precise. Simply put, it sets the initial condition in the recursion. Before someone says "there is a blue eyed person here", the initial condition in the recursion does not hold. The islanders are not allowed to say it so the visitor was the first one to make the statement. If indeed one of the islanders said it (whether blue or brown eyed) the condition would hold as well and the recursion would follow too. However none of them can talk about it.
- vacri 11y ago
  The problem is the initial leap of logic - why should any tribe member think that the foreigner is talking about them at all? The foreigner says 'there are other blue-eyed people' and everyone can see that there are indeed other blue-eyed people. There's no reason for anyone to think that they might be in the blue group. If there was some reason to start a count, sure, but there isn't. The foreigner just states that there are other blue-eyed people, which the tribe already knows.
  - orangecat 11y ago
    Reduce the problem to 2 blue-eyed people and it should be more clear. If Alice and Bob are the only people with blue eyes, then yes, Alice might think that the speaker was only talking about Bob. But when Bob fails to sacrifice himself the next day, that must be because he also sees a person with blue eyes, and that can only be Alice.
    - vacri 11y ago
      It's different with only two people than it is with multiple. When there's only one other person and they don't act the next day, then the logic works. But when there are numerous people and there is no reason to start counting, then it doesn't extrapolate out. Put more clearly: Foreigner says "blue eyes!". n is 2. If you have blue eyes and you don't see the other blue-eyed person top themselves, you conclude that you also have blue eyes as does the other person, and it's belly-opening time. But when n is 100, that initiation is missing. No-one tops themselves the next day? Hardly suprising - there are a hundred of the buggers. There is no reason to start counting, no "hang on, why didn't ol' blue eyes over there kick the bucket?". And, given that these are humans and not cronjobs, no-one is going to be patiently waiting day in, day out for 100 days to see when they'll all kill themselves. Drama can't even fester, since the tribesfolk are forbidden from talking about the issue. Life goes on, and over three months later, the foreigner's statement wouldn't cause a bloodbath :) TL;DR: if you want to make a fancy logic puzzle, don't use 'humans with bizarre rules' as your subjects. :)
      - knughit 11y ago
NhanH 11y ago
The new piece of information is that you also know everyone in the tribe know about that information. That was not a given before the statement.
- awesomepantsm 11y ago
  In the original, you know the other islanders can see a minimum of 98 BP (blue eyed people). And you know that they know that you know that they can see at least 97 BP. Given that, the statement that there is at least 1 blue eyed person doesn't provide new info. The argument that this "synchronizes" them is a little strange. For all we knew, they were in the middle of counting to 100 days prior to this statement due to the aforementioned common knowledge that everyone already knew there were blue eyed people.
  - alwaysmetara 11y ago
    They wouldn't be counting to 100 days, because there's no point where everyone would start counting together (no one is allowed to reveal anything about eye color). So you know there's no reason for all the blue eyed people to commit suicide together, so you can't reason anything out about the total number of blue eyed people.
  - awesomepantsm 11y ago
    As per a comment in the original article, they could choose to "sync" when one of their children is first told the rules about not knowing your own eye color. This changes the way that person would behave, and is a completely reasonable time for each islander to reevaluate the logical cascade.
    - jack9 11y ago
      > they could choose to "sync" They can't discuss it. That would be a difficult social experiment, that would risk your life.
- thedufer 11y ago
  Close, but not quite. As others have pointed out, that's not new information either. The information given is the knowledge that everyone knows that everyone knows...that at least one person has blue eyes, to infinite depth. The key is that this knowledge was one level short of allowing the chain reaction to start prior to the announcement.
aetherson 11y ago
This comment is going to be long, so buckle in. A few different ways of thinking about this issue: 1. I note that you say that when N = 3, there's no new information given. But when N = 2, all islanders are still aware that there is at least one person with blue eyes. I'm guessing that despite that fact, you see how in the N = 2 case, the two people with blue eyes can infer they have blue eyes. So the leap of intuition you should've had there is that somehow, new information is being conveyed, even though every islander already knows that there is at least one person with blue eyes. 2. But let's just tackle this with straightforward inductive logic. For now, we aren't going to worry about what new information is given, we're just going to do induction. If I see that N people have blue eyes, then I know that either there are N people with blue eyes, or there are N+1 people with blue eyes, and I have blue eyes. If after N days, the people I see with blue eyes haven't killed themselves, then I know that there must be N+1 people with blue eyes, and I have blue eyes, ergo I must kill myself. This holds with N=0 and N=1 and N=2, so it on some level doesn't matter what the logic is, induction works. 3. But okay, let's figure out what information is actually given. Let's say that on a 100 person island, I have blue eyes and Alice and Bob also have blue eyes. So from my perspective, I know that there are two possible scenarios: a. Alice, Bob, and I have blue eyes. b. Only Alice and Bob have blue eyes. Let's consider situation b. In situation b, what does Alice think? Since we're stipulating that I have brown eyes, Alice sees that Bob has blue eyes and that's it. Therefor, Alice thinks that there are two subscenarios of situation b: b1: Alice and Bob have blue eyes. b2: Only Bob has blue eyes. Hypothetical Alice in hypothetical situation b considers situation b2. She thinks about what Bob would know in situation b2. He would see that nobody besides potentially him has blue eyes. So he would kill himself when the visitor explained that at least one person had blue eyes. If Bob does not kill himself, then hypothetical Alice in situation b knows that she is not in situation b2. There is only one other subscenario of situation b, and that is b1, in which both Alice and Bob have blue eyes. So Alice would kill herself on day 2 (as would Bob). If Alice does not do that, then Alice clearly did not consider herself to be in situation b. The only alternative to situation b is situation a, in which Alice, Bob, and I have blue eyes. So we all kill ourselves on day 3. And obviously since all of this is symmetrical -- neither Alice, Bob, or I have any privileged information, that means that if I can see three people with blue eyes, and they don't kill themselves on day 3, then the only other valid scenario is that I also have blue eyes, so on day 4, I kill myself. The information added is information about what we know that other islanders know about people's eyes, not what we know about people's eyes.

alwaysmetara 11y ago

The important point is that the outsider's comment "synchronizes" the islander's information. On Day 2, say you are a blue eyed person. You can see two other blue eyed people. Since these two people don't commit suicide during that day, you know there must be three blue eyed people with one of them being you. So now you know your eye color.

deleted 11y ago
[deleted]
jack9 11y ago
> you know there must be three blue eyed people Why? The visitor didn't synchronize anything. All of the villagers already knew there was at least 1 blue eyed person. Per the story, 99 or 100 depending on your perspective.
- alwaysmetara 11y ago
  Everyone commits suicide at noon in view of everyone else. So the lowest possible number of blue eyed people go up one per day. Here's a longer explanation: Since people who know their eye color suicides at noon in plain sight of everyone else, we can set up induction. If there's only one blue eyed person, they would suicide the first day at noon. Since everyone sees > 1 blue eyed people, they know this wouldn't happen. Since this doesn't happen, it's common knowledge that there is more than 1 blue eyed person. So if you only see one other blue eyed person, you must have blue eyes (but you don't know this yet because you see two blue eyed people). On the second day, if there are only two blue eyed people, they would both suicide at noon. You see two blue eyed people. No one commits suicide because they either see two blue eyed people (if they have blue eyes) or three blue eyed people (if they don't). Now, everyone knows that there are more than two blue eyed people. Since no one committed suicide the day before, you reason that there must be three blue eyed people, with the 2 people you see plus yourself. Thus, you know your eye color. This is a bit long but I hope it's clear.
  - chii 11y ago
    the argument i hear about induction is that it's only valid IF n=1 is a true situation, n=2 is a true situation, etc, up to n=k (where k is 100 here). But we are only given n=100, and we cannot say that n=1,2,3... is true. The best counter example is the surprise execution paradox: suppose you are a prisoner, about to be executed next week. However, a strange rule applies - if you are not surprised by the date of your execution, you get to go free. You reason that you cannot be executed on sunday, the last day, since by friday, you'd know, and thus not be surprised. by induction, friday is also not a possible day, since you'd know by thurs, etc. You conclude that you cannot be surprised, and this will not get executed. But come wed, you get executed, and is completely surprised!
    - knughit 11y ago
      This problem is different from the surprise paradox, which relies on probability changing each day, and a possibility that either (a) the executioner will be proven a liar by randomly guessing Sunday, or (b) you confidently guessing Wednesday and being right.
  - jack9 11y ago
    > Now, everyone knows that there are more than two blue eyed people. That's incorrect, in a subtle way. The villagers (every single one) knew that before. It isn't discovered knowledge. Everyone (including the visitor) knows there are 99 or 100 because they can count them already, prior to the visitor. So nobody is suiciding for 99 days. As an individual who sees 99 other blue eyes, why would you think or not think that you have blue eyes on the next day?
    - alwaysmetara 11y ago
      I meant strictly more, as in at least three. If you know that there are more than two blue eyed people and you can only see two, you know that you have blue eyes. If you can see 99 blue eyes, you know that all the other people see either 98 or 99 (if there's a total of 99 blue eyes) or 99 or 100 (if there's 100 blue eyes). However, if some people see 98 and know that there's at least 99 people with blue eyes, then they would all suicide. Since they don't, you know everyone sees either 99 or 100 blue eyes. Since you see 99, you must have blue eyes.
awesomepantsm 11y ago
Why can't they synchronize at any other moment? They already knew N is either 99 or 100 even before the visitor came.
- millstone 11y ago
  Consider the case of 2 blue eyed people, Ben and Betty. Everybody knows there is at least one blue eyed person. But does everybody know that? That is, does everybody know that (everybody knows there is one blue eyed person)? From Ben's perspective, Betty might be the only blue-eyed person, so Ben does not know if Betty knows there is at least one blue eyed person. This is important because Ben needs to correlate Betty's actions with her knowledge; he can't do that if he doesn't know what her knowledge is. After the traveler's announcement, Ben knows that Betty knows there is at least one blue eyed person. This is the new information. And with three people, we add another layer: Ben does not know if Betty knows that Bobby has blue eyes. And so on. The visitor's announcement is special because it is infinitely layered. Everyone knows one person has blue eyes, and everyone knows that everyone knows, and so on, ad infinitum. This is called common knowledge: https://en.wikipedia.org/wiki/Common_knowledge_(logic) https://en.wikipedia.org/wiki/Common_knowledge_(logic)). Though to be fair, this isn't explicitly stated: the traveler "addresses the entire tribe" but it's critical that everyone in the tribe knows this is common knowledge. Merely addressing the entire tribe is not enough: if the traveler's announcement was in the form of a BCC mass mailing, there would be no new information and so no suicides.
  - knughit 11y ago
    It isn't infinitely layered! It is N-layered, which is critical.
- daxfohl 11y ago
  "synchronize" is a bit vague. It can be made more precise. Simply put, it sets the initial condition in the recursion. Before someone says "there is a blue eyed person here", the initial condition in the recursion does not hold. The islanders are not allowed to say it so the visitor was the first one to make the statement. If indeed one of the islanders said it (whether blue or brown eyed) the condition would hold as well and the recursion would follow too. However none of them can talk about it.
vacri 11y ago
The problem is the initial leap of logic - why should any tribe member think that the foreigner is talking about them at all? The foreigner says 'there are other blue-eyed people' and everyone can see that there are indeed other blue-eyed people. There's no reason for anyone to think that they might be in the blue group. If there was some reason to start a count, sure, but there isn't. The foreigner just states that there are other blue-eyed people, which the tribe already knows.
- orangecat 11y ago
  Reduce the problem to 2 blue-eyed people and it should be more clear. If Alice and Bob are the only people with blue eyes, then yes, Alice might think that the speaker was only talking about Bob. But when Bob fails to sacrifice himself the next day, that must be because he also sees a person with blue eyes, and that can only be Alice.
  - vacri 11y ago
    It's different with only two people than it is with multiple. When there's only one other person and they don't act the next day, then the logic works. But when there are numerous people and there is no reason to start counting, then it doesn't extrapolate out. Put more clearly: Foreigner says "blue eyes!". n is 2. If you have blue eyes and you don't see the other blue-eyed person top themselves, you conclude that you also have blue eyes as does the other person, and it's belly-opening time. But when n is 100, that initiation is missing. No-one tops themselves the next day? Hardly suprising - there are a hundred of the buggers. There is no reason to start counting, no "hang on, why didn't ol' blue eyes over there kick the bucket?". And, given that these are humans and not cronjobs, no-one is going to be patiently waiting day in, day out for 100 days to see when they'll all kill themselves. Drama can't even fester, since the tribesfolk are forbidden from talking about the issue. Life goes on, and over three months later, the foreigner's statement wouldn't cause a bloodbath :) TL;DR: if you want to make a fancy logic puzzle, don't use 'humans with bizarre rules' as your subjects. :)
    - knughit 11y ago

NhanH 11y ago

The new piece of information is that you also know everyone in the tribe know about that information. That was not a given before the statement.

awesomepantsm 11y ago
In the original, you know the other islanders can see a minimum of 98 BP (blue eyed people). And you know that they know that you know that they can see at least 97 BP. Given that, the statement that there is at least 1 blue eyed person doesn't provide new info. The argument that this "synchronizes" them is a little strange. For all we knew, they were in the middle of counting to 100 days prior to this statement due to the aforementioned common knowledge that everyone already knew there were blue eyed people.
- alwaysmetara 11y ago
  They wouldn't be counting to 100 days, because there's no point where everyone would start counting together (no one is allowed to reveal anything about eye color). So you know there's no reason for all the blue eyed people to commit suicide together, so you can't reason anything out about the total number of blue eyed people.
- awesomepantsm 11y ago
  As per a comment in the original article, they could choose to "sync" when one of their children is first told the rules about not knowing your own eye color. This changes the way that person would behave, and is a completely reasonable time for each islander to reevaluate the logical cascade.
  - jack9 11y ago
    > they could choose to "sync" They can't discuss it. That would be a difficult social experiment, that would risk your life.
thedufer 11y ago
Close, but not quite. As others have pointed out, that's not new information either. The information given is the knowledge that everyone knows that everyone knows...that at least one person has blue eyes, to infinite depth. The key is that this knowledge was one level short of allowing the chain reaction to start prior to the announcement.

aetherson 11y ago

This comment is going to be long, so buckle in. A few different ways of thinking about this issue: 1. I note that you say that when N = 3, there's no new information given. But when N = 2, all islanders are still aware that there is at least one person with blue eyes. I'm guessing that despite that fact, you see how in the N = 2 case, the two people with blue eyes can infer they have blue eyes. So the leap of intuition you should've had there is that somehow, new information is being conveyed, even though every islander already knows that there is at least one person with blue eyes. 2. But let's just tackle this with straightforward inductive logic. For now, we aren't going to worry about what new information is given, we're just going to do induction. If I see that N people have blue eyes, then I know that either there are N people with blue eyes, or there are N+1 people with blue eyes, and I have blue eyes. If after N days, the people I see with blue eyes haven't killed themselves, then I know that there must be N+1 people with blue eyes, and I have blue eyes, ergo I must kill myself. This holds with N=0 and N=1 and N=2, so it on some level doesn't matter what the logic is, induction works. 3. But okay, let's figure out what information is actually given. Let's say that on a 100 person island, I have blue eyes and Alice and Bob also have blue eyes. So from my perspective, I know that there are two possible scenarios: a. Alice, Bob, and I have blue eyes. b. Only Alice and Bob have blue eyes. Let's consider situation b. In situation b, what does Alice think? Since we're stipulating that I have brown eyes, Alice sees that Bob has blue eyes and that's it. Therefor, Alice thinks that there are two subscenarios of situation b: b1: Alice and Bob have blue eyes. b2: Only Bob has blue eyes. Hypothetical Alice in hypothetical situation b considers situation b2. She thinks about what Bob would know in situation b2. He would see that nobody besides potentially him has blue eyes. So he would kill himself when the visitor explained that at least one person had blue eyes. If Bob does not kill himself, then hypothetical Alice in situation b knows that she is not in situation b2. There is only one other subscenario of situation b, and that is b1, in which both Alice and Bob have blue eyes. So Alice would kill herself on day 2 (as would Bob). If Alice does not do that, then Alice clearly did not consider herself to be in situation b. The only alternative to situation b is situation a, in which Alice, Bob, and I have blue eyes. So we all kill ourselves on day 3. And obviously since all of this is symmetrical -- neither Alice, Bob, or I have any privileged information, that means that if I can see three people with blue eyes, and they don't kill themselves on day 3, then the only other valid scenario is that I also have blue eyes, so on day 4, I kill myself. The information added is information about what we know that other islanders know about people's eyes, not what we know about people's eyes.